Even with these conceptual and numerical advantages in many applications, stochastic methods cannot guarantee an increase in performance in general. This is because for complex systems, the benefit of only having to propagate wavefunctions rather than density matrices is outweighed by the cost of requiring large sample sizes to ensure `weak' convergence (which is to say, convergence at the ensemble level, rather than convergence at the level of a single unravelling, or so-called `strong' convergence) [1]}. Although various powerful schemes for reducing necessary sample sizes exist in specific applications of stochastic evolutions [2]}[3]}, there is no known general method for reducing the inherent statistical error below that guaranteed by the central limit theorem. For many applications of stochastic quantum propagation, this is not a problem, as the value in such methods often lies in (but is not limited to) the reproduction of realistic evolutions of a system of interest, for example. However, for the description of effects such as decoherence, a well-converged density matrix or equivalent is necessary, and in this respect, stochastic methods may not be able to out-perform the master equation formalism in general.
